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October 17th, 2016, 08:18 PM   #1
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[ASK] Probabilities

There are two boxes. Box I contains 5 red balls and 4 white balls. Box II contains 3 red balls and 6 white balls. A ball is taken randomly from box I and put into box II. Then, from box II, two balls are taken randomly. What is the probability of both balls taken from box II are red?

I only know that the probability of getting red ball from box I is $\displaystyle \frac{1}{9}$. How to do the rest?
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October 17th, 2016, 08:53 PM   #2
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The initial state is

$(5,4),~(3,6)$

If a red ball is chosen and transferred the state becomes

$(4,4),~(4,6)$

There is a probability of $\dfrac 5 9$ that this occurs

On the other hand if a white ball is chosen and transferred the state becomes

$(5,3),~(3,7)$

There is a probability of $\dfrac 4 9$ that this occurs.

From $(4,4),~(4,6)$ the probability of choosing 2 red balls from box 2 is

$\dfrac{4}{10} \cdot \dfrac{3}{9}$

From $(5,3),~(3,7)$ the probability of choosing 2 red balls from box 2 is

$\dfrac{3}{10}\cdot \dfrac{2}{9}$

So the total probability of selecting 2 balls from box 2 is

$\dfrac 5 9\cdot \dfrac{4}{10}\cdot \dfrac{3}{9} + \dfrac 4 9 \cdot \dfrac{3}{10}\cdot \dfrac{2}{9} =\dfrac{14}{135} $
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October 17th, 2016, 09:00 PM   #3
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Ugh... It's kinda hard to keep up with the explanation but I think I get it.
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